Research Article
N-prime Ideal Graph of a Finite Commutative Ring
Kalamani Duraisamy
,
Hariharan Murugan*
Issue:
Volume 15, Issue 6, December 2026
Pages:
220-232
Received:
30 September 2026
Accepted:
12 October 2026
Published:
5 November 2026
Abstract: In this paper, we introduce and investigate the N-prime ideal graph of a finite commutative ring with unity. The concept of prime ideal graphs over rings has been studied by many authors. Let R be a finite commutative ring with unity and let J be a proper ideal of R. The N-prime ideal graph of R with respect to J, denoted by GJ, is the simple undirected graph whose vertex set consists of the elements of the ring R and the two distinct vertices are adjacent if and only if their product belongs to that ideal and at least one of them belongs to the sum of that ideal and the nilradical of the ring. The aim of this article is to study the structural properties of the N-prime ideal graph and establish the connection between the algebraic properties of ideals and their graph theoretic properties.In particular, the N-prime ideal graphs are characterized as star, pineapple, and complete multipartite graphs. The subgraphs induced by nilpotent and idempotent elements are investigated,along with their connections to zero-divisor and product maximal graphs. Furthermore, the properties of the N-prime ideal graph as a cograph and a threshold graph are studied and illustrated with examples. The study demonstrates how N-prime ideal properties can be effectively represented and analyzed using graph theoretic methods.
Abstract: In this paper, we introduce and investigate the N-prime ideal graph of a finite commutative ring with unity. The concept of prime ideal graphs over rings has been studied by many authors. Let R be a finite commutative ring with unity and let J be a proper ideal of R. The N-prime ideal graph of R with respect to J, denoted by GJ, is the simple undir...
Show More
Research Article
A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances
Fatemeh Sharafi*
Issue:
Volume 15, Issue 6, December 2026
Pages:
233-243
Received:
20 September 2026
Accepted:
4 October 2026
Published:
14 November 2026
DOI:
10.11648/j.acm.20261506.12
Downloads:
Views:
Abstract: This paper presents an advanced analysis of the radial basis function (RBF) collocation method for solving time-dependent partial differential equations (PDEs). A unified framework is developed that integrates weighted finite difference temporal discretization with global RBF spatial approximation for both smooth and piecewise smooth radial kernels. We provide a rigorous stability analysis of the resulting time-stepping scheme and systematically examine the roles of the shape parameter, polynomial augmentation, and scattered point distribution in controlling accuracy and conditioning. In addition to the classical global collocation formulation, we survey and compare contemporary localized and high-order RBF variants—including localized RBF collocation (LRBF), RBF partition of unity (RBF-PU), ghost-point-enhanced RBF pseudo-spectral (RBF-PS-G), and RBF finite difference (RBF-FD)—that have emerged as the state of the art for large-scale and long-time simulations in 2024–2026. Numerical experiments on benchmark problems including the telegraph and Klein–Gordon equations, a nonlinear viscous Burgers equation, a three-dimensional Helmholtz problem, and comparisons between global RBF, RBF-PU, and LRBF validate the theoretical predictions. The method's simplicity, mesh-free nature, and high-order accuracy make it a competitive and highly adaptable tool for a broad class of PDE problems.
Abstract: This paper presents an advanced analysis of the radial basis function (RBF) collocation method for solving time-dependent partial differential equations (PDEs). A unified framework is developed that integrates weighted finite difference temporal discretization with global RBF spatial approximation for both smooth and piecewise smooth radial kernels...
Show More
